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Journal of Mathematical Physics

Quantum theory and measures on Hilbert space

Henry Krips

Published May 1, 1977• 1 citation
Publisher

Abstract

It is shown that merely the decision to represent the fundamental events of quantum theory by vectors in a Hilvert space uniquely determines the form of any probability measure over the set of events. The proof proceeds by showing that any nonnegative meausre on the vectors of a Hilbert space, for which the sum of the measures on the vectors of any complete orthonormal set is 1, takes the usual form damanded by quantum theory. This result represents a considerable strengthening of one of the consequences of Gleason’s theorem.

DOI
10.1063/1.523382

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It is shown that merely the decision to represent the fundamental events of quantum theory by vectors in a Hilvert space uniquely determines the form of any probability measure over the set of events. The proof proceeds by showing that any nonnegative meausre on the vectors of a Hilbert space, for which the sum of the measures on the vectors of any complete orthonormal set is 1, takes the usual form damanded by quantum theory. This result represents a considerable strengthening of one of the consequences of Gleason’s theorem.
The postulates of quantum theory are rather abstract in comparison with those of other physical theories such as special relativity. This thesis considers two tools for investigating this discrepancy and makes a connection between them. The first of these tools, Gleason-type theorems, illustrates the interplay between postulates concerning observables, states and probabilities of measurement outcomes, demonstrating that they need not be entirely independent. Gleason’s original and remarkable result applied to observables described by projection-valued measures; however, the theorem does not hold in dimension two. Busch generalised the idea to observables described by positive operator measures, proving a result which holds for all separable Hilbert spaces. We show that Busch’s assumptions may be weakened without affecting the result. The manner in which we weaken the assumptions brings them closer to Gleason’s original treatment of projection-valued measures. We will then demonstrate the connection between Gleason-type theorems and Cauchy’s functional equation, a connection which yields an alternative proof of Busch’s result. The second tool we consider is the family of general probabilistic theories which offers a means of comparing quantum theory with reasonable alternatives. We identify a general probabilistic theory which reproduces the set of non-local correlations achievable in quantum theory, a property often thought to be particular to quantum theory. Finally, we connect these two tools by determining the class of general probabilistic theories which admit a Gleason-type theorem.
In quantum physics the tests of most properties do not have predetermined outcomes. The latter have nevertheless well-defined probabilities of being realized during a test. Following Popper we interpret these probabilities as physical propensities. A first purpose of the present article is to formalize the propensity interpretation in the framework of state-property structures. Next, Gleason’s theorem asserts that in the Hilbert space there exists a unique propensity function (i.e., one probability measure for each state vector); the propensities are thus uniquely determined by the state vector. Conversely, we prove that if the state-property structure admits one and only one propensity function, then the set ℒ of all properties is a complete atomic orthomodular lattice. We point out that according to our assumption the probabilistic aspect of the system is entirely determined by its deterministic aspect. Assuming furthermore that each property can be ideally tested, it follows that ℒ is isomorphic to the direct union of Hilbertian space lattices. We recover thus the purely classical and purely quantum frameworks as the two extreme cases. The intermediate cases correspond to quantum mechanics with—possibly continuous—superselection variables. Finally, we prove that a system is classical, i.e., all properties are mutually compatible, if and only if the propensity function is dispersion free. In our approach the quantum probabilities appear thus as a generalization of classical determinism rather than a generalization of classical probabilities.
In the quantum probability theory the 6-field of random events is replaoed by the lattice of orthogonal projectors in a separable infinite dimensional Hilbert space H.A countably additive function from this lattices to the unit interval constitutes a state, the non-commutative analogue of a probability measure.The Theorem of Gleason [4] asserts that every state is of the form n--tr jtT, where jt runs over all projectors and T is a probability operator on H, i.e. a positive linear operator of unit trace.Conversely, every probability operator determines a state by the Gleason formula.From now onwards let S stand for the set of all states, i.e. all probability operators on H.We shall dienote by the space of-? ell nuclear linear operators acting in H with the norm jjïj|.j= tr{TT*) 1 ^2.Of coarse S is e closed and convex subset Of TyIn quantum theory to every physical quantity or observable th^re corresponds e sslf»adjoint not necessarily bounded linear operator on Hs By 0 ws shall denote the set of all obt ;ry£bles-Given he-0 f tac pi'cb^ivility distribution of A at the etato T is defined for ail Eoi'oi subsets E of the real
1. Introduction.In quantum field theory we must consider the Hilbert space having non-countable bases which corresponds to a sequence of non-negative integers (n, n.,...).Since we can construct one-to-one mapping from the set of the sequences (n, n,...) onto the points in 0, 1 interval [8, we can identify these bases to [0, 1 interval.Let 7 denote a point in 0, 1 interval and let r be the element of the Hilbert space which corresponds to y.The element of this Hilbert space is usually represented by the formulae flCrrdt( 7) and Cr, in [3], E4] and [6], where C, Cr are constants, and d/(') is a measure on E0, 1].By single dz(r), however, we cannot represent every element of this Hilbert space.That is to say, by a continuous measure dz('), we cannot represent the element of the second form.On the other hand by the second form, we cannot represent the element of the first form.